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Parallel Coordinates

Visual Multidimensional Geometry and Its Applications

verfasst von: Alfred Inselberg

Verlag: Springer New York

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The breakthrough idea of parallel coordinates has enchanted many people with its cleverness and power. Al Inselberg, the source of that cleverness and power, ?nally shares the depth and breadth of his invention in this historically important book. I’ve been waiting impatiently for this exposition since I was captivated by Inselberg’slectureattheUniversityofMarylandin1979.Hehadalreadyunderstood the potential parallel coordinates has for generating insights into high-dimensional analysis and information visualization. In the following decades he polished the arguments, built effective software, and demonstrated value in important appli- tions. Nowabroadcommunityofreaderscanbene?tfromhisinsightsandeffective presentation. I believe that Inselberg’s parallel coordinates is a transformational ideal that ´ matches the importance of Rene Descartes’ (1596–1650) invention of Cartesian coordinates. Just as Cartesian coordinates help us understand 2D and 3D geometry, parallel coordinates offer fresh ways of thinking about and proving theorems in higher-dimensional geometries. At the same time they will lead to more powerful tools for solving practical problems in a wide variety of applications. It is rare to encounter such a mind-shattering idea with such historic importance. While Inselberg’s insight and exposition opens the door to many discoveries, there is much work to be done for generations of mathematicians, computer sci- tists, programmers, and domain experts who will need to build on these innovative ideas.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

About half of our sensory neurons are dedicated to vision,⁵ endowing us with a remarkable pattern-recognition ability. Visualization, an emerging field with huge potential, aims to incorporate this tremendous capability into our problem-solving process. Rather than dictionary definitions we consider visualization as a collection of application-specific mappings.

Alfred Inselberg

2. Geometry Background

Abstract

The Nile’s flooding in Egypt used to destroy landmarks and position markers,raising the need for reliable measurements and originating the development of geometry Geo-Гεω(for land) metry (for measure-μετρια).¹⁰ The Rhind Papyrus, which is in the British Museum, was written by an Egyptian priest named Ahmes about 1,800 B.C.E. It treats the measurement of planar regions and solids.

Alfred Inselberg

3. ♣ FT-1 The Plane with Parallel Coordinates

Abstract

A point in the x ₁ x ₂ plane is represented in parallel coordinates (abbreviated ∥-coords) by a line in the xy-plane (Fig. 3.1). To find out what a line “looks like” in ∥-coords, a set of collinear points is selected (Fig. 3.2, top right), and the lines representing them intersect (top left) at a point!

Alfred Inselberg

4. Multidimensional Lines

Abstract

There are several point ↔line dualities (such as the very useful Hough transform) in the plane that do not generalize well for higher dimensions. This is because the natural duality is point ↔hyperplane in projective N-space ℙ^N with point ↔line when N = 2. However, in ∥-coords there is a useful and direct generalization, which is the subject of this chapter. At first the basic idea for lines in ℝ^N, rather than ℙ^N, is derived intuitively, paving the way for the general case, which is treated subsequently (as in [106] and [107]).

Alfred Inselberg

5. Planes, p-Flats, and Hyperplanes

Abstract

A hyperplane in ℝ^Ncan be translated to one that contains the origin, which is an (N−1)-dimensional linear subspace of ℝ^N. Since ℝ^N ⁻ ¹ can be represented in ∥ -coords by N−1 vertical lines and a polygonal line representing the origin, it is reasonable to expect a similar representation for hyperplanes in ℝ^N.

Alfred Inselberg

6. ** Envelopes

Abstract

Envelopes are encountered in many areas of mathematics as well as in several applications. Also, they are useful background for the construction of representations in parallel coordinates. In our era, envelopes are no longer a “fashionable” subject and are often not covered in coursework. For good references one needs to look at older sources. For these reasons, a quick overview is included here.

Alfred Inselberg

7. Curves

Abstract

The image of each point P on a curve c is a line obtained from (7.1), as shown in Fig. 7.1 (left), and the envelope of all such lines, when it exists, is the curve

Alfred Inselberg

8. Proximity of Lines, Planes, and Flats

Abstract

In order to apply the results of the representation of flats by indexed points, their behavior in the presence of errors needs to be understood. This was briefly touched on in Chapter 5, where we saw that small variations in the coefficients of a plane’s equation yield a pair of closely spaced point clusters representing the corresponding “close” planes. The encouraging message is that “closeness” is visually evident. To pursue this in depth, we need some concepts discussed in Chapter 7.

Alfred Inselberg

9. Hypersurfaces in ℝ N

Abstract

Early in the development of ∥-coords, the representation of surfaces was attempted by simply plotting the surface’s points as polygonal lines. Except for very special surfaces(i.e.,spheres,ellipsoids),thisturnedouttobeevenmorehopelessthanwhat happened with line-curves. The next improvement was the surface representation by the envelopeof these polygonal lines.

Alfred Inselberg

10. ♣ FT Data Mining and Other Applications

Abstract

The first, and still most popular, application of parallel coordinates is in exploratory data analysis (EDA) to discover data subsets (relations) that fulfill certain objectives and guide the formulation of hypotheses. A data set with Mitems has 2^Msubsets, any one of which may be the one we really want. With a good data display, our fantastic pattern-recognition ability cannot only cut great swaths in our search through this combinatorial explosion, but also extract insights from the visual patterns. These are the core reasons for data visualization.

Alfred Inselberg

11. Recent Results

Abstract

This chapter suggests ways to improve the visualization of several lines in parallel coordinates, specifically how to overcome the inconvenience caused by the fact that the indexed points are not necessarily arranged “neatly” along the horizontal axis. Several approaches are presented by which the input data (or alternatively, the axes) may be transformed so as to reduce or eliminate the problem, with minimal information loss.

Alfred Inselberg

12. Solutions to Selected Exercises

Abstract

Draw a circle with any center Band any radius r. Choose a point Aon the circle. Draw a circle with center Aof radius r. Both circles intersect at C. With Cas the center draw an arc on circle Bto obtain the point D. With Das the center draw an arc to obtain the point E. The points A,B,Eare collinear.

Alfred Inselberg

Backmatter

Titel: Parallel Coordinates
verfasst von: Alfred Inselberg
Verlag: Springer New York
Electronic ISBN: 978-0-387-68628-8
Print ISBN: 978-0-387-21507-5
DOI: https://doi.org/10.1007/978-0-387-68628-8